Siciak's homogeneous extremal functions, holomorphic extension and a generalization of Helgason's support theorem

2019/02/04 by Bergh, Jöran, Sigurdsson, Ragnar
#32U3 #Complex Variables (math.CV) #FOS: Mathematics #Primary 32A15 #Secondary 30D15

paper · doi:10.48550/arxiv.1902.01134

Abstract

We prove that a function, which is defined on a union of lines ℂ E through the origin in ℂⁿ with direction vectors in E⊂ ℂⁿ and is holomorphic of fixed finite order and finite type along each line, extends to an entire holomorphic function on ℂⁿ of the same order and finite type, provided that E has positive homogeneous capacity in the sense of Siciak and all directional derivatives along the lines satisfy a necessary compatibility condition at the origin. We are able to estimate the indicator function of the extension in terms of Siciak's weighted homogeneous extremal function, where the weight is a function of the type of the given function on each given line. As an application we prove a generalization of Helgason's support theorem by showing how the support of a continuous function with rapid decrease at infinity can be located from partial information on the support of its Radon transform.

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